Resonance, Peaking, and Bandwidth Relationships

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Core Idea

Resonance occurs when system natural frequency aligns with input frequency, causing amplitude amplification above DC gain. Peak resonance magnitude (Mr, resonance peak) and resonant frequency (ωr) depend on damping: lower damping yields higher peaks and sharpened resonance. The relationship between Mr, bandwidth, and damping provides design insight: reducing damping increases bandwidth but increases peaking and overshoot—a fundamental design trade-off.

Explainer

You know from bandwidth and cutoff frequency analysis that a system's frequency response has a characteristic shape — flat at low frequencies, then rolling off. You also know from second-order system theory that the natural frequency ω_n and damping ratio ζ together define how a system responds. Resonance is the phenomenon that connects these: when the driving frequency is close to ω_n, the system amplifies the input rather than attenuating it — the output is *larger* than the input, not smaller.

The physical mechanism is energy exchange. An underdamped second-order system stores energy in two forms (think of a spring-mass system: kinetic and potential, or an LC circuit: magnetic and electric). Near the natural frequency, energy sloshes back and forth between the two storage elements in synchrony with the driving signal. If damping is low, little energy escapes each cycle, and the oscillation grows large. At exactly the resonant frequency ω_r = ω_n √(1 − 2ζ²), the magnitude of the frequency response reaches its peak M_r = 1 / (2ζ√(1 − ζ²)). Notice that as ζ → 0, M_r → ∞ — an undamped system driven at resonance grows without bound.

The damping ratio governs a direct trade-off between three related quantities: peaking, bandwidth, and time-domain overshoot. Reducing ζ (less damping) increases M_r (more peaking in frequency domain), increases bandwidth (the −3 dB frequency rises), and increases percent overshoot in the step response. These are not independent consequences you can pick among — they are manifestations of the same underlying system pole locations moving closer to the imaginary axis. A system with ζ = 0.707 ("critically flat" or Butterworth response) has M_r ≈ 1 (no peaking) and about 4% step overshoot — often a good starting point for design. A system with ζ = 0.5 has M_r ≈ 1.15 and about 16% overshoot.

For control design, this trade-off is a central constraint. You can make a system respond faster (wider bandwidth, lower ζ), but you pay with peaking and overshoot — the system overshoots its target and may oscillate before settling. You can make a system well-damped (high ζ, low overshoot), but it becomes sluggish and slow to reject disturbances. Real specifications typically constrain both: a step response must settle within X% of final value in time T, *and* peak no more than Y% above — translating directly into constraints on ζ and ω_n. Understanding the resonance-damping-bandwidth relationship is what lets you read those specs and immediately reason about whether they are achievable and at what cost.

Practice Questions 5 questions

Prerequisite Chain

Counting to 10 → Counting to 20 → Understanding Zero → The Number Zero → Counting to Five → One-to-One Correspondence → Combining Small Groups Within 5 → Addition Within 10 → Addition Within 20 → Two-Digit Addition Without Regrouping → Two-Digit Addition with Regrouping → Addition Within 100 → Repeated Addition as Multiplication → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Two-Digit by One-Digit Division → Division with Remainders → Remainders and Quotients in Division → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Reading and Writing Decimals → Comparing and Ordering Decimals → Adding and Subtracting Decimals → Multiplying Decimals → Dividing Decimals → Dividing Fractions → Mixed Number Arithmetic → Order of Operations → Integer Order of Operations → Variable Expressions → Combining Like Terms → One-Step Equations → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals over Rectangular Regions → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Green's Theorem → Surface Integrals and Flux of Vector Fields → Surface Integrals and Flux of Vector Fields → Divergence Theorem: Flux and Outflow → Divergence Theorem → Electric Flux → Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → Second-Order Transient Circuit Response → Feedback Control Fundamentals → Laplace Transform Methods for Control → Transfer Functions and System Modeling → Poles, Zeros, and System Stability → Characteristic Equation and Closed-Loop Stability → Natural Frequency and Damping Ratio → Resonance, Peaking, and Bandwidth Relationships

Longest path: 113 steps · 629 total prerequisite topics

Prerequisites (3)

Bandwidth and Frequency Domain Specificationshard Natural Frequency and Damping Ratiosoft Bandwidth and Resonant Frequency Selectionsoft

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